Question: The length of the day in Lagos, Nigeria (in minutes), $t$ days since the beginning of the year can be estimated by this formula: $ L(t) = 727+22\sin \left(\dfrac{2\pi (t - 80.75)}{365}\right)$. When is the longest day of the year? Give an exact answer.
Explanation: The longest day of the year occurs when $\sin\left(\dfrac{2\pi (t - 80.75)}{365}\right)$ reaches its maximum. $\sin u$ is largest when $u$ is $\dfrac{\pi}{2}$ plus a multiple of $2\pi$. So the longest day of the year is when $\dfrac{2\pi (t - 80.75)}{365} = \dfrac{\pi}{2} + 2\pi n$ for $n$, an integer. We can solve that equation for $t$ : $\begin{aligned}t - 80.75 &= 91.25 + 365 n \\ t &= 172 + 365 n. \end{aligned}$ Since changing the value of $n$ just adds a multiple of $365$ days, it just changes what year we're in: in each year, the longest day of the year is $172$ days since its beginning. The longest day of the year in Lagos, Nigeria, is $172$ days since its beginning.